Since energy dissipation assumes a key role in the TB-TT transition, it is crucial to account for the membrane viscosity in order to give a realistic description of the erythrocyte dynamics. Even though several works explore the RBC TB-TT transition as related to the shear rate \(\dot{\gamma }\)  and the viscosity ratio \(\lambda \)  (see  [1]}, [2]}, [3]}, [4]} and references therein), only few works focus on the effect of membrane viscosity \(\mu _{\mbox{\scriptsize m}}\) . This paper aims at taking some steps further to fill this gap. In our previous work [5]}, the effect of membrane viscosity during the relaxation after the cessation of a mechanical load was investigated; here, the quantitative effect of the membrane viscosity on the TB-TT transition is explored, adopting the same numerical model as in [5]}. More specifically, the Skalak model [7]} and the Helfrich formulation [8]} are implemented to describe the elastic behaviour, while the Standard Linear Solid model is employed to account for the membrane viscosity [9]}; the fluid is resolved in the framework of the Lattice Boltzmann Method (LBM) [10]}.
